关于L内射覆盖

We use the class of L-injective modules to define L-injective covers, and provide the characterizations of L-injective covers by the properties of kernels of homomorphisms. We prove that the right L-noetherian right L-hereditary ring is just such that every right R-module has an L-injective cover which is monic. We also use kernels of homomorphisms to investigate L-simple L-injective covers and give some constructions of L-simple L-injective covers.     

Rings Whose Modules Have Grade Zero

In this paper, we prove that R is a two-sided Artinian ring and J is a right annihilator ideal if and only if (i) for any nonzero right module, there is a nonzero linear map from it to a projective module; (ii) every submodule of RR is not a radical module for some right coherent rings. We call a ring a right X ring if Homa(M, R) = 0 for any right module M implies that M = 0. We can prove some left Goldie and right X rings are right Artinian rings. Moreover we characterize semisimple rings by using X rings. A famous Faith‘s conjecture is whether a semipimary PF ring is a QF ring. Similarly we study the relationship between X rings and QF and get many interesting results.     

Inductive Rings and Systems of Diophantine Equations

In this paper, by using model-theoretic methods, it is shown that some systems of unsolved cubic diophantine equations in number theory can have solutions in certain inductive extension rings of the ring I of rational integers. These inductive rings are not fields, and every element of them is a sum of 4 cubes and a sum of 3 squares. Also some of them satisfy the Goldbach conjecture and some others don＇t.     

正则环和GP-内射环

A ring R is called left Gp-injective if for any a∈R, there exists a positive integer n such that any left R-homomorphism of Raⁿ into R extends to one of R into R. In this paper, we prove that the centre of semiprime (left nonsingular) left GP- injective ring is regular ring, and improve some propositions in [3].     

一些部分差集和相对差集的构造

In this paper, we give a construction of RDS in Galois ring by using some bent function, and obtain the equivalent relationship between RDS and a kind of bent function. At the same time, its existence is demonstrated.     

对几类相关环的讨论

In this paper,the relations between a quasi-perfect ring,a coherent ring and a π-coherent ring are discussed.It is proved that a quasi-perfect ring is Morita equivalent.After that a moer popular ting,i.e,FGFT-ring,is discussed,some interesting properties are obtained.     

弱对偶环

In This paper, the concept of weakly dual ring is introduced, which is a proper generalization of the dual ring. If R is a right weakly dual ring, then (1) Z(RR) = J(R); (2) If R is also a zero-division power ring, then R is a right AP-injective ring. In addition, some properties of weakly dual rings are given.     

Generalized PP and Zip Subrings of Matrix Rings

Let R be an abelian ring. We consider a special subring An, relative to α2,…, αn∈ REnd（R）, of the matrix ring Mn（R） over a ring R. It is shown that the ring An is a generalized right PP-ring （right zip ring） if and only if the ring R is a generalized right PP-ring （right zip ring）. Our results yield more examples of generalized right PP-rings and right ziu rings.     

Structure of Two-Groups Associative Rings

Introduction.T.S.Ravisanker and U.S.Shukla [1] introduced the notion of a Weak ring A, more general than a ring and a ring in the sense of Nobusawa, and obtained analogical characterizations of the Jacobson radical for weak rings. It is clear that every ring A is a weak ring A for some abelian group (Theorem 3.1 in [2]. Therefore auther gives consideration to such a fact that A and are equal in the weak ring, moreover there is     

Some criteria for the Cohen-Macaulay property and local cohomology

Let a be an ideal of a commutative Noetherian ring R and M be a finitely generated R-module of dimension d. We characterize Cohen-Macaulay rings in term of a special homological dimension. Lastly, we prove that if R is a complete local ring, then the Matlis dual of top local cohomology module Ha^d（M） is a Cohen-Macaulay R-module provided that the R-module M satisfies some conditions.     

Strong commutativity-preserving generalized derivations on semiprime rings

Let R be a ring with a subset S. A mapping of R into itself is called strong commutativitypreserving （scp） on S, if [f（x）, f（y）] = [x, y] for all x, y ∈ S. The main purpose of this paper is to describe the structure of the generalized derivations which are scp on some ideals and right ideals of a prime ring, respectively. The semiprime case is also considered.     

关于超素环的左经典商环

In this paper,we define the left rank of a ring R and discuss the relatlons between a left ore reng R and its left classical quotient ring Q on the properties of their ideals and left ranks.Also we gave four equivalent types of the proposition:a ring R has a left classical quotient ring Q and Q is a division ring.As an application of this result,we gave a brief proof method which different from [2] on the theorem of the Goldie Prime rings.     

LIE IDEALS, MORITA CONTEXT AND GENERALIZED （α β）-DERIVATIONS

A classical problem in ring theory is to study conditions under which a ring is forced to become commutative. Stimulated from Jacobson＇s famous result, several tech- niques are developed to achieve this goal. In the present note, we use a pair of rings, which are the ingredients of a Morita context, and obtain that if one of the ring is prime with the generalized （α β）-derivations that satisfy certain conditions on the trace ideal of the ring, which by default is a Lie ideal, and the other ring is reduced, then the trace ideal of the reduced ring is contained in the center of the ring. As an outcome, in case of a semi-projective Morita context, the reduced ring becomes commutative.     

A Note on Ideals of a Three-Dimensional Associative Algebra

In this paper, we shall apply the theory of nilpotent radical to derive some results about ideals of a there-dimensional associative algebra over a field F and of an associative ring of order p~3 where p is a prime. These results are useful in studying the structure of the algebra and the ring.     

线性变换的GM元对

In this short note we discuss the GM property of some special linear transformation pairs over infinite-dimensional vector spaces. In particular, it is proved that if R = End(VD)is the endomorphism ring of an infinite-dimensional right vector space V over a division ring D with |C(D)| 3 and g ∈ R, then(a0+ a1g, g) is a GM pair for any a0, a1∈ C(D).Furthermore, two existing results are obtained as immediate consequences.     

Lie Homomorphism on the Associative Ring

Is it true tliat Lie homomorpHisin (isomorphism) \phi of a ring R into a ring R' is a homomorphism (isomorpliism) ? Herstein,I. N. and Kleinfeld, E. and Martindal, W. S.. obtained some results for simple ring and primitive ring,. In this paper I shall study the Lie homomorphism on the associative ring and arrive at the following main conclusion: Theorem. Suppose that R,R' are both associative rings with 1,and the center ofR’ does not contain zero divisor, where R’ is not of oharaoteristio 2 or 3. If \phi is a 3-Lie homomorphism of R onto R',then \phi must be either a homoinorpiiisin or the negative of an anti-homomorphism of R onto R'. Theorem. Suppose that R is an associatiye ring and $R'\ne {0}$ is a prime ring,where (R', +) does not contain elements of the period 2 or 3. If \phi is a 3-Lie homomorphism of R onto R',then \phi it either a homomorphism or the negative of an antihomomorpliism of R onto R'. Theorem. Suppose that R is an associative ring and (R, +) does not contain element of the period 2 or 3,R' is a prime ring. If \phi is a 3-Lie isomorpJlism of R onto R'。then \phi is either a isomorphism or the negative of an anti-isomorphism of R onto R'.     

Standardness and Standard Automorphisms of Chevalley Groups, I: the Case of Rank at Least Two

The author studies the linkage between the standardness and the standard automorphisms of Chevalley groups over rings.It is proved that if H is any standard subgroup of G(R),then each of its automorphisms can be extended to an automorphism of G(R,I),restricted to an automorphism of E(R,I),and an automorphism of E(R,I) can be extended to one of G(R,I).The case of Chevalley groups of rank at least two is treated in this paper.Further results about the case of Chevalley groups of rank one,the case of non-commutative ground ring and some others exceptions will appear elsewhere.     

斜Armendariz矩阵环

Let R be a ring.We show in the paper that the subring Un(R) of the upper triangular matrix ring Tn(R) is α-skew Armendariz if and only if R is α-rigid,also it is maximal in some non α-skew Armendariz rings,where α is a ring endomorphism of R with α(1) = 1.     

On（N,U）-Coherence of Modules and Rings

Let U be a flat right R-module and N an infinite cardinal number.A left R-module M is said to be (N,U)-coherent if every finitely generated submodule of every finitely generated M-projective module in σ[M] is (N,U)-finitely presented in σ[M].It is proved under some additional conditions that a left R-module M is (N,U)-coherent if and only if Л^Ni∈I U is M-flat as a right R-module if and only if the (N,U)-coherent dimension of M is equal to zero.We also give some characterizations of left (N,U)-coherent dimension of rings and show that the left N-coherent dimension of a ring R is the supremum of (N,U)-coherent dimensions of R for all flat right R-modules U.     

Zip模(英文)

A ring R is called right zip provided that if the annihilator τR（X） of a subset X of R is zero, then τR（Y） = 0 for some finite subset Y C X. Such rings have been studied in literature. For a right R-module M, we introduce the notion of a zip module, which is a generalization of the right zip ring. A number of properties of this sort of modules are established, and the equivalent conditions of the right zip ring R are given. Moreover, the zip properties of matrices and polynomials over a module M are studied.